Quadratic + cubic residue => 6th power residue?

by Miquel-point, May 16, 2025, 6:02 PM

Decide whether the following statement is true: if an infinite arithmetic sequence of positive integers includes both a perfect square and a perfect cube, then it also includes a perfect $6$ th power.

Proposed by Sándor Róka, Nyíregyháza

number theory

Cute property of Pascal hexagon config

by Miquel-point, May 16, 2025, 5:59 PM

In cyclic hexagon $ABCDEF$ let $P$ denote the intersection of diagonals $AD$ and $CF$ , and let $Q$ denote the intersection of diagonals $AE$ and $BF$ . Prove that if $BC=CP$ and $DP=DE$ , then $PQ$ bisects angle $BQE$ .

Proposed by Géza Kós, Budapest

geometry

Cyclic Hexagon

II_a - r_a = R - r implies A = 60

by Miquel-point, May 16, 2025, 5:55 PM

The incenter and the inradius of the acute triangle $ABC$ are $I$ and $r$ , respectively. The excenter and exradius relative to vertex $A$ is $I_a$ and $r_a$ , respectively. Let $R$ denote the circumradius. Prove that if $II_a=r_a+R-r$ , then $\angle BAC=60^\circ$ .

Proposed by Class 2024C of Fazekas M. Gyak. Ált. Isk. és Gimn., Budapest

geometry

Anything real in this system must be integer

by Assassino9931, May 9, 2025, 9:26 AM

Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
$\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}$ and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.

Marek Maruin, Slovakia

This post has been edited 1 time. Last edited by Assassino9931, May 9, 2025, 9:26 AM

algebra

Al-Khwarizmi IJMO

Gives typical russian combinatorics vibes

by Sadigly, May 8, 2025, 4:15 PM

You are given a positive integer $n$ . $n^2$ amount of people stand on coordinates $(x;y)$ where $x,y\in\{0;1;2;...;n-1\}$ . Every person got a water cup and two people are considered to be neighbour if the distance between them is $1$ . At the first minute, the person standing on coordinates $(0;0)$ got $1$ litres of water, and the other $n^2-1$ people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.

Prove that, no matter what, the person standing on the coordinates $(x;y)$ will not have more than $\frac1{x+y+1}$ litres of water.

This post has been edited 2 times. Last edited by Sadigly, May 11, 2025, 6:40 AM

combinatorics

AZE SENIOR NMO

Good Permutations in Modulo n

by swynca, Apr 27, 2025, 2:03 PM

An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$ , such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$ ;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$ .
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

This post has been edited 2 times. Last edited by swynca, Apr 27, 2025, 4:15 PM

BMO

number theory

Concurrency from isogonal Mittenpunkt configuration

by MarkBcc168, Apr 28, 2020, 7:07 AM

Let $\triangle ABC$ be a scalene triangle with circumcenter $O$ , incenter $I$ , and incircle $\omega$ . Let $\omega$ touch the sides $\overline{BC}$ , $\overline{CA}$ , and $\overline{AB}$ at points $D$ , $E$ , and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$ . The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$ . The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $\text{[math]}$ , $FP$ , and $OI$ are concurrent.

Proposed by MarkBcc168.

This post has been edited 1 time. Last edited by MarkBcc168, Apr 28, 2020, 7:08 AM

geometry

Triangular Numbers in action

by integrated_JRC, Oct 7, 2018, 11:26 AM

Find all natural numbers $n$ such that $1+[\sqrt{2n}]~$ divides $2n$ .

( For any real number $x$ , $[x]$ denotes the largest integer not exceeding $x$ . )

This post has been edited 3 times. Last edited by integrated_JRC, Oct 8, 2018, 1:16 AM

algebra

number theory

Divisibility

Number theory problem

by Angelaangie, Jun 19, 2018, 4:51 PM

Prove that 7p+3^p-4 it is not a perfect square where p is prime.

number theory

JBMO Shortlist

JBMO

another n x n table problem.

by pohoatza, May 13, 2007, 11:38 AM

Consider a $n$ x $n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$ x $2$ square which contains an odd number of unit squares white colored.

modular arithmetic

induction

combinatorics proposed

combinatorics

Submit

Check out the Pascal's Law post. I included a cartoon from the xkcd serial webcomic.
by aoum, Yesterday at 1:04 AM
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100 posts!
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The first few posts for April are out!
by aoum, Apr 1, 2025, 11:51 PM
Sure! I understand that it would be quite a bit to take in.
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No, but it is a lot to take in. Also, could you do the Gamma Function next?
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Yes, I'll be doing problems of the day every day.
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I think it would also be cool if you did a problem of the day every day, as I see from today's problem.
by jocaleby1, Mar 5, 2025, 1:13 AM
Do you guys like my "lectures" or would you like something else?
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Yeah, keep on making these "lectures"
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Anyone wants to contribute to my blog? Shout or give me a friend request!
by aoum, Mar 2, 2025, 3:22 PM
Nice blog! Contrib?
by jocaleby1, Mar 1, 2025, 6:43 PM

60 shouts

Pi in the Sky

by Miquel-point, May 16, 2025, 6:02 PM

by Miquel-point, May 16, 2025, 5:59 PM

by Miquel-point, May 16, 2025, 5:55 PM

by Assassino9931, May 9, 2025, 9:26 AM

by Sadigly, May 8, 2025, 4:15 PM

by swynca, Apr 27, 2025, 2:03 PM

by MarkBcc168, Apr 28, 2020, 7:07 AM

by integrated_JRC, Oct 7, 2018, 11:26 AM

by Angelaangie, Jun 19, 2018, 4:51 PM

by pohoatza, May 13, 2007, 11:38 AM